I haven’t found anything online to suggest that the two actresses are related; Wiki is silent on the point. It’s not all that common a name in the U.S., though.
Wiki notes that Polly Bergen’s birth name is Nellie Paulina Burgin. Different spelling.
Ventriloquist/comic Edgar Bergen was Candice’s dad, right?
I believe that’s correct.
It was either him or Charlie McCarthy, can’t recall which. 
Or maybe Mortimer Snerd?